A Combinatorial Approach to Involution and Delta-Regularity I: Involutive Bases in Polynomial Algebras of Solvable Type

Reference: Preprint, Universität Mannheim, 2002

Description: Finally, the new version of this paper is finished. In fact, it has not much to do with the old version... Large parts have been completely rewritten. Furthermore, it is now split into two parts. This first part discusses the basic theory of involutive bases. I treat at once the theory within a rather general class of non-commutative algebras: the polynomial algebras of solvable type (note that I generalise the definition given by Kandry-Rodi and Weispfenning). This class includes besides the ordinary commutative polynomials e.g. rings of linear differential or difference operators, universal enveloping algebras and some quantum algebras. As we must distinguish between left and right ideals in a non-commutative setting, I also study the construction of involutive bases for two-sided ideals. Two further new aspects are involutive bases with respect to arbitrary semigroup orders, as they appear in local computations, and involutive bases in polynomial algebras over rings. In the former case I analyse besides Lazard's approach (treated for the special case of the Weyl algebra already in [ Weyl ]) also the approach via Mora's normal form. In the latter case some assumptions must be made on the commutation relations in the polynomial algebra for the involutive completion algorithm to terminate. In both cases, in general only weak involutive bases exist. This new concept is introduced here for the first time: such weak bases are still Grbner bases but they do not define a disjoint decomposition of the ideal.

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